In Exercises 41–58, find dy/dt.


y = (t⁻³/⁴ sin(t))⁴/³

Verified step by step guidance

1. First, identify the function y in terms of t: \( y = \left( t^{-\frac{3}{4}} \sin(t) \right)^{\frac{4}{3}} \). We need to find \( \frac{dy}{dt} \).
2. Apply the chain rule to differentiate \( y \) with respect to \( t \). The chain rule states that if \( y = u^n \), then \( \frac{dy}{dt} = n u^{n-1} \frac{du}{dt} \). Here, \( u = t^{-\frac{3}{4}} \sin(t) \) and \( n = \frac{4}{3} \).
3. Differentiate \( u = t^{-\frac{3}{4}} \sin(t) \) with respect to \( t \) using the product rule. The product rule states that if \( u = v \cdot w \), then \( \frac{du}{dt} = v' \cdot w + v \cdot w' \). Here, \( v = t^{-\frac{3}{4}} \) and \( w = \sin(t) \).
4. Calculate \( v' \) and \( w' \): \( v' = \frac{d}{dt}(t^{-\frac{3}{4}}) = -\frac{3}{4}t^{-\frac{7}{4}} \) and \( w' = \frac{d}{dt}(\sin(t)) = \cos(t) \). Substitute these into the product rule to find \( \frac{du}{dt} \).
5. Substitute \( u \), \( n \), and \( \frac{du}{dt} \) back into the chain rule expression to find \( \frac{dy}{dt} = \frac{4}{3} \left( t^{-\frac{3}{4}} \sin(t) \right)^{\frac{1}{3}} \cdot \frac{du}{dt} \). Simplify the expression to complete the differentiation process.

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(t)) is composed of two functions, then its derivative dy/dt is f'(g(t)) * g'(t). This rule is essential for finding the derivative of y = (t⁻³/⁴ sin(t))⁴/³, as it involves nested functions.

Product Rule

The product rule is used to differentiate functions that are products of two or more functions. If y = u(t) * v(t), then the derivative dy/dt is u'(t) * v(t) + u(t) * v'(t). In the given problem, y = t⁻³/⁴ * sin(t) is a product of two functions, requiring the application of the product rule to find its derivative.

Power Rule

The power rule is a basic differentiation rule used when differentiating functions of the form y = t^n. It states that the derivative dy/dt is n * t^(n-1). In the expression y = (t⁻³/⁴ sin(t))⁴/³, the power rule is applied to the outer function to help find the derivative of the entire expression.